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Half Opended Infinite Intervals
We now define a second kind of improper integral where the interval is infinite. DEFINITION Let f be continuous on the halfopen interval [a, ∞). The improper integral of f from a to ∞ is defined by the limit The improper integral is said to converge if the limit exists and to diverge otherwise. Here is a description of this kind of improper integral using definite integrals with hyperreal endpoints. Let f be continuous on [a, ∞). (1) (if and only if for all positive infinite H. (2) = ∞ (or ∞) if and only if is positive infinite (or negative infinite) for all positive infinite H.
The last two examples give an unexpected result. A region with infinite area is rotated about the xaxis and generates a solid with finite volume! In terms of hyperreal numbers, the area of the region under the curve y = x^{2/3} from 1 to an infinite hyperreal number H is equal to 3(H^{1/3}  1), which is positive infinite. But the volume of the solid of revolution from 1 to H is equal to 3π(1  H^{1/3}), which is finite and has standard part 3n. We can give a simpler example of this phenomenon. Let if be a positive infinite hyperinteger, and form a cylinder of radius l/H and length H^{2} (Figure 6.7.6). Then the cylinder is formed by rotating a rectangle of length H^{2}, width 1/H, and infinite area H^{2}/H = H. But the volume of the cylinder is equal to n, V = πr^{2}h = π(1/H)^{2}(H^{2}) = π. Figure 6.7.6: Area = H, volume = ir Imagine a cylinder made out of modelling clay, with initial length and radius one. The volume is π. The clay is carefully stretched so that the cylinder gets longer and thinner. The volume stays the same, but the area of the cross section keeps getting bigger. When the length becomes infinite, the cylinder of clay still has finite volume V = π, but the area of the cross section has become infinite.


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